US7427521B2 - Generating simulated diffraction signals for two-dimensional structures - Google Patents

Generating simulated diffraction signals for two-dimensional structures Download PDF

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US7427521B2
US7427521B2 US10/274,252 US27425202A US7427521B2 US 7427521 B2 US7427521 B2 US 7427521B2 US 27425202 A US27425202 A US 27425202A US 7427521 B2 US7427521 B2 US 7427521B2
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profile
hypothetical
diffraction signal
section
simulated diffraction
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US20040078173A1 (en
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Joerg Bischoff
Xinhui Niu
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Tokyo Electron Ltd
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TEL Timbre Technologies Inc
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Priority to CNB200380101531XA priority patent/CN100442067C/zh
Priority to KR1020057006261A priority patent/KR101058476B1/ko
Priority to PCT/US2003/032779 priority patent/WO2004036142A2/en
Priority to AU2003279290A priority patent/AU2003279290A1/en
Priority to DE10393515T priority patent/DE10393515T5/de
Priority to JP2004545360A priority patent/JP4805579B2/ja
Priority to TW092128747A priority patent/TWI263911B/zh
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    • GPHYSICS
    • G06COMPUTING; CALCULATING OR COUNTING
    • G06FELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10Complex mathematical operations
    • GPHYSICS
    • G01MEASURING; TESTING
    • G01NINVESTIGATING OR ANALYSING MATERIALS BY DETERMINING THEIR CHEMICAL OR PHYSICAL PROPERTIES
    • G01N21/00Investigating or analysing materials by the use of optical means, i.e. using sub-millimetre waves, infrared, visible or ultraviolet light
    • G01N21/17Systems in which incident light is modified in accordance with the properties of the material investigated
    • G01N21/47Scattering, i.e. diffuse reflection
    • G01N21/4788Diffraction

Definitions

  • the present invention relates to wafer metrology, and more particularly to generating simulated diffraction signals for use in optical metrology.
  • Optical metrology can be utilized to determine the profile of structures formed on semiconductor wafers.
  • optical metrology involves directing an incident beam at a structure and measuring the resulting diffraction beam.
  • the characteristics of the measured diffraction beam i.e., a measured diffraction signal
  • pre-determined diffraction signals i.e., simulated diffraction signals
  • the profile associated with the matching simulated diffraction signal is presumed to represent the profile of the structure.
  • the process of generating a simulated diffraction signal involves performing a large number of complex calculations, which can be time and computationally intensive.
  • the number and complexity of the calculations increases for structures having profiles that vary in more than one dimension.
  • one or more simulated diffraction signals for use in determining the profile of a structure formed on a semiconductor wafer are generated, where the profile varies in more than one dimension.
  • Intermediate calculations are generated for variations in a hypothetical profile of the structure in a first dimension and a second dimension, where each intermediate calculation corresponds to a portion of the hypothetical profile of the structure.
  • the generated intermediate calculations are then stored and used in generating one or more simulated diffraction signals for one or more hypothetical profiles of the structure.
  • FIG. 1 depicts an exemplary optical-metrology system
  • FIGS. 2A-2E depict various hypothetical profiles of a structure
  • FIG. 3 depicts an exemplary one-dimension structure
  • FIG. 4 depicts an exemplary two-dimension structure
  • FIG. 5 is a top view of an exemplary structure
  • FIG. 6 is a top view of another exemplary structure
  • FIG. 7 depicts a chain of harmonic orders
  • FIG. 8 depicts an array of harmonic orders
  • FIG. 9 depicts an exemplary process of generating simulated diffraction signals
  • FIG. 10 depicts an exemplary cache of blocks of hypothetical layers and a library.
  • FIG. 11 depicts cause and response fields of the front and back of an exemplary layer or block of layers
  • FIGS. 12A and 12B depict an exemplary process of generating simulated diffraction signals for a hypothetical profile of a structure
  • FIGS. 13A and 13B depict another exemplary process of generating simulated diffraction signals for a hypothetical profile of a structure
  • FIGS. 14A and 14B depict still another exemplary process of generating simulated diffraction signals for a hypothetical profile of a structure
  • FIG. 15 depicts exemplary blocks of hypothetical layers assembled to correspond to exemplary hypothetical profiles of an exemplary two-dimension structure
  • FIG. 16 depicts an exemplary computer system.
  • optical-metrology can be utilized to determine the profile of a structure formed on a semiconductor wafer.
  • an optical-metrology system 100 can include an electromagnetic source 120 , such as an ellipsometer, reflectometer, and the like.
  • a structure 145 is illuminated by an incident beam 110 from electromagnetic source 120 .
  • Incident beam 110 is directed onto structure 145 at an angle of incidence ⁇ 1 with respect to normal ⁇ right arrow over (n) ⁇ of structure 145 .
  • Diffraction beam 115 leaves at an angle of ⁇ d with respect to normal ⁇ right arrow over (n) ⁇ .
  • diffraction beam 115 is received by detector 170 .
  • electromagnetic source 120 is an ellipsometer
  • the relative magnitude ( ⁇ ) and the phase ( ⁇ ) of diffraction beam 115 are received and detected.
  • electromagnetic source 120 is a reflectometer
  • the relative intensity of diffraction beam 115 is received and detected.
  • optical-metrology system 100 includes a processing module 190 , which converts diffraction beam 115 received by detector 170 into a diffraction signal (i.e., a measured diffraction signal). As described below, the profile of structure 145 can then be determined using either a library-based process or a regression-based process.
  • the measured diffraction signal is compared to a library of simulated diffraction signals. More specifically, each simulated diffraction signal in the library is associated with a hypothetical profile of the structure. When a match is made between the measured diffraction signal and one of the simulated diffraction signals in the library, such as within a preset criteria, the hypothetical profile associated with the matching simulated diffraction signal is presumed to represent the actual profile of the structure. The matching simulated diffraction signal and/or hypothetical profile can then be utilized to determine whether the structure has been fabricated according to specifications.
  • processing module 190 compares the measured diffraction signal to simulated diffraction signals stored in a library 185 .
  • Library 185 includes simulated diffraction signals that are associated with hypothetical profiles of structure 145 . More particularly, in one exemplary embodiment, library 185 includes pairings of simulated diffraction signals and hypothetical profiles of structure 145 . The simulated diffraction signal in each pairing includes hypothetically generated reflectances that characterize the predicted behavior of diffraction beam 115 assuming that the profile of the structure 145 is that of the hypothetical profile in the simulated diffraction signal and hypothetical profile pairing.
  • the set of hypothetical profiles stored in library 185 can be generated by characterizing a hypothetical profile using a set of parameters, then varying the set of parameters to generate hypothetical profiles of varying shapes and dimensions.
  • the process of characterizing a profile using a set of parameters can be referred to as parameterizing.
  • hypothetical profile 200 can be characterized by parameters h 1 and w 1 that define its height and width, respectively.
  • additional shapes and features of hypothetical profile 200 can be characterized by increasing the number of parameters.
  • hypothetical profile 200 can be characterized by parameters h 1 , w 1 , and w 2 that define its height, bottom width, and top width, respectively.
  • the width of hypothetical profile 200 can be referred to as the critical dimension (CD).
  • parameter wl and w 2 can be described as defining the bottom CD and top CD, respectively, of hypothetical profile 200 .
  • the set of hypothetical profiles stored in library 185 can be generated by varying the parameters that characterize the hypothetical profile. For example, with reference to FIG. 2B , by varying parameters h 1 , w 1 , and w 2 , hypothetical profiles of varying shapes and dimensions can be generated. Note that one, two, or all three parameters can be varied relative to one another.
  • the number of hypothetical profiles and corresponding simulated diffraction signals in the set of hypothetical profiles and simulated diffraction signals stored in library 185 depends, in part, on the range over which the set of parameters and the increment at which the set of parameters are varied.
  • the hypothetical profiles and the simulated diffraction signals stored in library 185 are generated prior to obtaining a measured diffraction signal from an actual structure.
  • the range and increment (i.e., the range and/or resolution) used in generating library 185 can be selected based on familiarity with the fabrication process for a structure and what the range of variance is likely to be.
  • the range and/or resolution of library 185 can also be selected based on empirical measures, such as measurements using AFM, X-SEM, and the like.
  • the measured diffraction signal is compared to a simulated diffraction signal (i.e., a trial spectrum signal).
  • the simulated diffraction signal is generated prior to the comparison using a set of parameters (i.e., trial parameters) for a hypothetical profile (i.e., a hypothetical profile). If the measured diffraction signal and the simulated diffraction signal do not match, such as within a preset criteria, another simulated diffraction signal is generated using another set of parameters for another hypothetical profile, then the measure-diffraction signal and the newly generated simulated diffraction signal are compared.
  • the hypothetical profile associated with the matching simulated diffraction signal is presumed to represent the actual profile of the structure.
  • the matching simulated diffraction signal and/or hypothetical profile can then be utilized to determine whether the structure has been fabricated according to specifications.
  • processing module 190 can generate a simulated diffraction signal for a hypothetical profile, then compare the measured diffraction signal to the simulated diffraction signal. As described above, if the measured diffraction signal and the simulated diffraction signal do not match, such as within a preset criteria, then processing module 190 can iteratively generate another simulated diffraction signal for another hypothetical profile.
  • the subsequently generated simulated diffraction signal can be generated using an optimization algorithm, such as global optimization techniques, which includes simulated annealing, and local optimization techniques, which includes steepest descent algorithm.
  • the simulated diffraction signals and hypothetical profiles can be stored in a library 185 (i.e., a dynamic library).
  • the simulated diffraction signals and hypothetical profiles stored in library 185 can then be subsequently used in matching the measured diffraction signal.
  • simulated diffraction signals are generated to be compared to measured diffraction signals.
  • simulated diffraction signals can be generated by applying Maxwell's equations and using a numerical analysis technique to solve Maxwell's equations. More particularly, in the exemplary embodiment described below, rigorous coupled-wave analysis (RCWA) is used. It should be noted, however, that various numerical analysis techniques, including variations of RCWA, can be used.
  • RCWA In general, RCWA involves dividing a profile into a number of sections, slices, or slabs (hereafter simply referred to as sections). For each section of the profile, a system of coupled differential equations generated using a Fourier expansion of Maxwell's equations (i.e., the components of the electromagnetic field and permittivity ( ⁇ )). The system of differential equations is then solved using a diagonalization procedure that involves eigenvalue and eigenvector decomposition (i.e., Eigen-decomposition) of the characteristic matrix of the related differential equation system. Finally, the solutions for each section of the profile are coupled using a recursive-coupling schema, such as a scattering matrix approach.
  • a recursive-coupling schema such as a scattering matrix approach.
  • the Fourier expansion of Maxwell's equations is obtained by applying the Laurent's rule or the inverse rule.
  • the rate of convergence can be increased by appropriately selecting between the Laurent's rule and the inverse rule. More specifically, when the two factors of a product between permittivity ( ⁇ ) and an electromagnetic field (E) have no concurrent jump discontinuities, then the Laurent's rule is applied. When the factors (i.e., the product between the permittivity ( ⁇ ) and the electromagnetic filed (E)) have only pairwise complimentary jump discontinuities, the inverse rule is applied.
  • a periodic grating depicted in FIG. 3 has a profile that varies in one dimension (i.e., the x-direction), and is assumed to be substantially uniform or continuous in the y-direction.
  • the Fourier expansion for the periodic grating depicted in FIG. 3 is performed only in the x direction, and the selection between applying the Laurent's rule and the inverse rule is also made only in the x direction.
  • a periodic grating depicted in FIG. 4 has a profile that varies in two dimensions (i.e., the x-direction and the y-direction).
  • the Fourier expansion for the periodic grating depicted in FIG. 4 is performed in the x direction and the y-direction, and the selection between applying the Laurent's rule and the inverse rule is also made in the x direction and the y direction.
  • Fourier expansion can be performed using an analytic Fourier transformation (e.g., a sin(v)/v function).
  • analytic Fourier transformation e.g., a sin(v)/v function.
  • Fourier expansion can be performed using an analytic Fourier transformation only when the structure has a rectangular patched pattern, such as that depicted in FIG. 5 .
  • a numerical Fourier transformation e.g., by means of a Fast Fourier Transformation
  • harmonic orders are placed in relation to each other.
  • this process is straightforward because the harmonic orders can be ordered in a chain.
  • harmonic orders ranging from ⁇ 2 to +2 can be ordered in a simple chain from ⁇ 2 to +2.
  • each harmonic order can be related to each other on a one-to-one basis.
  • the diffraction orders occur in an array.
  • the array of diffraction orders is projected into one dimension. Therefore, each pair of orders are sorted and the algorithm is organized correspondingly to ensure the correct orders are used.
  • the dispersion relation in order to compute the modes outside and inside a homogeneous section (i.e., Rayleigh modes for a section) and to formulate the differential equations for the electromagnetic field, the dispersion relation is defined as follows:
  • m and n are the number of the two-dimensional diffraction order related to x and y directions, respectively;
  • p x and p y are the respective periods;
  • k is the wave number (2 ⁇ / ⁇ ); and
  • is supposed to be a real number, i.e., there is no absorption.
  • ⁇ 2 + ⁇ 2 is less than k 2 , then ⁇ is purely a real number, which means physically that the respective mode is propagating. However, if ⁇ 2 + ⁇ 2 is greater than k 2 , ⁇ is purely an imaginary number, which means that the respective mode is evanescent, i.e., exponentially decaying. Therefore, the orders can be sorted related to the value of the term ⁇ 2 + ⁇ 2 starting with the least values. In this way, it is ensured that the ⁇ m,n (with m,n indicating the two-dimensional order) and thus the orders m,n itself are sorted by the degree of their evanescence.
  • the eigenvalues and eigenvectors can be obtained from the Eigen-decomposition.
  • the corresponding modes are called Bragg-modes.
  • the eigenvalues can be taken directly or the square root is computed. In the latter case, an appropriate solution of the square root can be obtained by enforcing that the imaginary part of the square root is positive. Then, the eigenvalues and thus the orders are sorted by the imaginary part starting with the least value, and the eigenvectors are sorted correspondingly.
  • a checkerboard grating is depicted in FIG. 4 as a two-dimension structure. It should be recognized, however, that a two-dimension structure can include various structures having profiles that vary in more than one dimension, such as contact holes, posts, and the like.
  • the process of generating a simulated diffraction signal can involve performing a large number of complex calculations. Additionally, as the complexity of the hypothetical profile increases, so does the number and complexity of the calculations needed to generate the simulated diffraction signal for the hypothetical profile.
  • a portion of the calculations performed in generating simulated diffraction signals can be stored as intermediate calculations prior to generating one or more simulated diffraction signals.
  • the system of differential equations for each section of a profile is solved using, in part, Eigen-decomposition.
  • the eigenvalues and eigenvectors i.e., the eigensolutions
  • the previously calculated and stored eigensolutions are retrieved when the system of differential equations are to be solved in generating one or more simulated diffraction signals.
  • step 902 prior to generating one or more simulated diffraction signals for one or more hypothetical profiles, a set of eigensolutions are calculated, which will be used in generating the one or more simulated diffraction signals for the one or more hypothetical profiles.
  • the set of eigensolutions are calculated for the various geometries and/or materials that may be used in generating the one or more simulated diffraction signals for the one or more hypothetical profiles.
  • each hypothetical profile is divided into a number of sections, where the geometry of a particular section corresponds to the geometry of a particular portion of the hypothetical profile.
  • each hypothetical profile can be comprised of layers of different material.
  • a particular section can correspond to the material in a particular portion of the hypothetical profile.
  • a hypothetical profile can be divided into a number of rectangular sections, with each rectangular section having a different width that corresponds to the width of a particular portion of the hypothetical profile.
  • the eigensolutions are sensitive to changes in width, meaning that rectangular sections of different width have different eigensolutions, they are not sensitive to changes in height, meaning that rectangular sections of different height have the same eigensolutions.
  • a set of eigensolutions can be calculated for the various width of rectangular sections that may be used in generating the library.
  • the hypothetical profile comprises a layer of oxide on top of a layer of resist.
  • the hypothetical profile can be divided into at least two sections with the first section corresponding to the oxide layer and the second section corresponding to the resist layer.
  • eigensolutions are calculated for the oxide layer and the resist layer.
  • eigensolutions are calculated for rectangular sections of varying widths that are comprised of oxide, which will correspond to the first section of the hypothetical profiles, and rectangular sections of varying widths that are comprised of resist, which will correspond to the second section of the hypothetical profiles.
  • the sections of a hypothetical profile have been depicted in two dimensions. It should be noted, however, that for a structure having a profile that varies in two or more dimensions (i.e., two-dimension structure), the sections can have more complex shapes and parameterizations.
  • the rectangular sections can have widths in two dimensions (e.g., a width in a x-direction, a width in a y-direction, which may be orthogonal or not), one or more rounded comers, and the like.
  • the elliptical sections can have diameters (e.g., a diameter in a x-direction, a diameter in a y-direction), a parameter that describes the deviation from an ellipse or a rectangle (e.g., an elliptical exponent that is equal to 2 for an ellipse and increases the more the ellipsoidal cross section shape approaches a rectangle), and the like.
  • diameters e.g., a diameter in a x-direction, a diameter in a y-direction
  • a parameter that describes the deviation from an ellipse or a rectangle e.g., an elliptical exponent that is equal to 2 for an ellipse and increases the more the ellipsoidal cross section shape approaches a rectangle
  • the sections can have various shapes of varying complexity.
  • the sections can include combinations of shapes, such as a combination of rectangular shapes.
  • step 904 the set of calculated eigensolutions are stored.
  • the set of calculated eigensolutions can be stored in memory, file, and the like. Additionally, the set of calculated eigensolutions can be indexed for ease of retrieval.
  • sets of calculated eigensolutions are generated and stored for varying geometry and/or material as well as varying wavelength or angle of incidence for use in generating one or more simulated diffraction signals based on the stored eigensolutions.
  • the sets of calculated eigensolutions are generated and stored in accordance with a nesting hierarchy, meaning that the eigensolutions are generated and stored in multiple nested iterations (i.e., loops) of one or more parameters (i.e., geometry, material, wavelength, angle of incidence, and the like).
  • first loop the geometry of the sections is varied.
  • first loop is repeated with the material of the sections varied for each iteration of the first loop.
  • first two iterations are repeated with the wavelength of the incident beam varied for each iteration of the first loop and each iteration of the second loop.
  • angle resolved metrology i.e., metrology where the angle of incidence is varied
  • the angle of incidence of the incident beam can be varied rather than the wavelength. It should be recognized, however, that the order and number of loops can vary.
  • a hypothetical profile is generated.
  • a hypothetical profile can be characterized using a set of parameters, which can then be varied to generate the set of hypothetical profiles.
  • step 908 the hypothetical profile is divided into sections.
  • step 910 for each section, a system of differential equations are generated.
  • step 912 the previously calculated and stored eigensolution that corresponds to the section are retrieved.
  • step 914 the system of differential equations are solved using the retrieved eigensolution for the section.
  • step 916 if the last section has not been reached, steps 910 to 914 are repeated for the remaining sections of the hypothetical profile. If the last section has been reached, in step 918 , the solutions for the sections of the hypothetical profiles are coupled to arrive at a solution for the hypothetical profile. In step 920 , this solution is then stored as the simulated diffraction signal for the hypothetical profile.
  • step 922 if a simulated diffraction signal for another hypothetical profile is to be generated, steps 908 to 920 are repeated. If the last hypothetical profile has been reached, the process is stopped.
  • a library of hypothetical profiles and corresponding simulated diffraction signals are generated and stored for varying geometry and/or material as well as varying wavelength or angle of incidence based on the previously generated and stored eigensolutions.
  • the library of hypothetical profiles and corresponding simulated diffraction signals are generated and stored in the same nesting hierarchy as the previously generated and stored eigensolutions.
  • one or more simulated diffraction signals are generated based on the previously generated and stored eigensolutions.
  • calculating and storing the eigensolutions in advance of generating one or more simulated diffraction signals can reduce the computation time for generating the one or more simulated diffraction signals.
  • the computation time for generating one or more simulated diffraction signals is further reduced by generating intermediate calculations (hereafter referred to as “diffraction calculations”) for a plurality of blocks of sections (hereafter referred to as “hypothetical layers”).
  • a plurality of hypothetical layers 1002 can be grouped together as a block of hypothetical layers 1004 (e.g., block of hypothetical layers 1004 . 1 - 1004 . 4 ).
  • Each hypothetical layer 1002 i.e., a section characterizes a layer (i.e., a portion) of a hypothetical profile.
  • hypothetical layers 1002 . 1 , 1002 . 2 , and 1002 . 3 are depicted as being grouped together to form block of hypothetical layers 1004 . 1 .
  • Hypothetical layers 1002 . 4 , 1002 . 5 , and 1002 . 6 are depicted as being grouped together to form block of hypothetical layers 1004 . 2 .
  • Hypothetical layers 1002 . 7 , 1002 . 8 , and 1002 . 9 are depicted as being grouped together to form block of hypothetical layers 1004 . 3 .
  • Hypothetical layers 1002 . 10 , 1002 . 11 , and 1002 . 12 are depicted as being grouped together to form block of hypothetical layers 1004 . 4 .
  • blocks of hypothetical layers 1004 of various sizes and shapes can be generated. Although blocks of hypothetical layers 1004 are depicted in FIG. 10 as including three hypothetical layers 1002 , it should be recognized that they can include any number of hypothetical layers 1002 .
  • Diffraction calculations are generated for each hypothetical layer 1002 within a block of hypothetical layers 1004 .
  • Diffraction calculations for each block of hypothetical layers 1004 are then generated by aggregating the diffraction calculations for each hypothetical layer 1002 within each block of hypothetical layers 1004 .
  • the results of the diffraction calculations are scattering matrices.
  • a scattering matrix S connects the cause to the response fields of the front and back of a layer or block of layers.
  • the scattering matrix has 4 sub-matrices (i.e., reflection matrix r f and the transmission matrix t f at front side excitation as well as the reflection matrix r b and the transmission matrix t b at back side excitation):
  • pairs of diffraction calculations and blocks of hypothetical layers 1004 are stored in a cache 1006 .
  • diffraction calculations are generated for hypothetical layers 1002 . 1 , 1002 . 2 , and 1002 . 3 .
  • a diffraction calculation for block of hypothetical layers 1004 . 1 is then generated by aggregating diffraction calculations for hypothetical layers 1002 . 1 , 1002 . 2 , and 1002 . 3 .
  • Block of hypothetical layers 1004 . 1 and the diffraction calculations associated with block of hypothetical layers 1004 . 1 are then stored in cache 1006 .
  • diffraction calculations for blocks of hypothetical layers 1004 . 2 , 1004 . 3 , and 1004 . 4 are generated and stored in cache 1006 .
  • cache 1006 may contain tens and hundreds of thousands of blocks of hypothetical layers 1004 and diffraction calculations.
  • FIG. 10 depicts blocks of hypothetical layers 1004 being stored in cache 1006 in a graphical format
  • blocks of hypothetical layers 1004 can be stored in various formats.
  • blocks of hypothetical layers 1004 can be stored using parameters that define their shapes.
  • the stored blocks of hypothetical layers 1004 can then be used to generate one or more simulated diffraction signals for one or more hypothetical profiles. More specifically, one or more blocks of hypothetical layers 1004 can be used to characterize a hypothetical profile, then the corresponding diffraction calculations of the one or more blocks of hypothetical layers 1004 can be used to generate the simulated diffraction signal for the hypothetical profile.
  • one block of hypothetical layers 1004 can be used to characterize a hypothetical profile. For example, with reference to FIG. 12A , assume for the sake of this example that a simulated diffraction signal is to be generated for hypothetical profile 1200 A. As described above, a block of hypothetical layers 1004 from cache 1006 is selected that characterizes hypothetical profile 1200 A. The appropriate block of hypothetical layers 1004 can be selected using an error minimization algorithm, such as a sum-of-squares algorithm.
  • block of hypothetical layers 1004 . 1 is selected from cache 1006 .
  • the diffraction calculation associated with block of hypothetical layers 1004 . 1 is then retrieved from cache 1006 .
  • Boundary conditions are then applied to generate the simulated diffraction signal for hypothetical profile 1200 A.
  • the parameters that define hypothetical profile 1200 A can be varied to define another hypothetical profile.
  • the parameters are varied to define hypothetical profile 1200 B.
  • block of hypothetical layers 1004 . 4 is selected from cache 1006 .
  • the diffraction calculation associated with block of hypothetical layers 1004 . 4 is then retrieved from cache 1006 .
  • Boundary conditions are then applied to generate the simulated diffraction signal for hypothetical profile 1200 B.
  • multiple blocks of hypothetical layers 1004 can be used to characterize a hypothetical profile.
  • profile 1300 A can be characterized by blocks of hypothetical layers 1004 . 1 , 1004 . 2 , and 1004 . 3 .
  • an error minimization algorithm can be utilized to determine the appropriate blocks of hypothetical layers 1004 to use in characterizing profile 1300 A.
  • profile 1300 A the diffraction calculations associated with blocks of hypothetical layers 1004 . 1 , 1004 . 2 , and 1004 . 3 are retrieved from cache 1006 . Boundary conditions are then applied to generate the simulated diffraction signal for profile 1300 A. More particularly, the boundary conditions at the top and bottom of profile 1300 A are applied.
  • the parameters that define hypothetical profile 1300 A can be varied to define another profile.
  • the parameters are varied to define hypothetical profile 1300 B.
  • hypothetical profile 1300 B can be characterized by blocks of hypothetical layers 1004 . 1 , 1004 . 3 , and 1004 . 4 .
  • the diffraction calculations associated with blocks of hypothetical layers 1004 . 1 , 1004 . 3 , and 1004 . 4 are retrieved from cache 1006 .
  • Boundary conditions are then applied to generate the simulated diffraction signal for hypothetical profile 1300 B.
  • a hypothetical profile can include multiple materials.
  • hypothetical profile 1400 is used to characterize an actual profile of a grating periodic formed from two materials. More particularly, as depicted in FIG. 14A , hypothetical profile 1400 includes a first layer 1402 and a second layer 1404 .
  • first layer 1402 represents an oxide layer of the actual profile
  • second layer 1404 represents a metal layer of the actual profile.
  • hypothetical profile 1400 can include any number of layers to represent layers of any number of materials in an actual profile.
  • cache 1006 includes blocks of hypothetical layers 1004 of various materials as well as various shapes.
  • blocks of hypothetical profiles 1004 . 1 and 1004 . 3 represent oxide layers
  • hypothetical profiles 1004 . 2 and 1004 . 4 represent metal layers.
  • profile 1400 can be characterized by blocks of hypothetical layers 1004 . 3 and 1004 . 4 from cache 1006 . Boundary conditions are then applied to generate the simulated diffraction signal for profile 1400 .
  • hypothetical layers 1002 and blocks of hypothetical layers 1004 have been depicted as having rectangular and trapezoidal shapes, respectively.
  • the diffraction calculation for a hypothetical layer 1002 depends on its width but not on its height.
  • the diffraction calculation for a block of hypothetical layers 1004 depends on its height as well as its width. Therefore, the blocks of hypothetical layers 1004 stored in cache 1006 are characterized and indexed, in part, by their width and their height. More particularly, when blocks of hypothetical layers 1004 have symmetric-trapezoidal shapes, they can be indexed by their height, bottom width (bottom CD), and top width (top CD).
  • blocks of hypothetical layers 1004 can have various shapes. As such, they can be characterized and indexed using any number of parameters.
  • a hypothetical profile has been described and depicted as being characterized by a combination of sections (as described in section 3) and a combination of blocks of layers (as described here in section 4).
  • a hypothetical profile can be characterizing using a combination of one or more blocks of hypothetical layers and one or more sections.
  • hypothetical layers and blocks of hypothetical layers have been depicted in two dimensions. It should be noted, however, that for a structure having a profile that varies in two or more dimensions (i.e., two-dimension structure), the hypothetical layers and/or blocks of hypothetical layers can have more complex shapes and parameterizations.
  • the blocks can have widths in two dimensions (e.g., a width in a x-direction, a width in a y-direction, which may be orthogonal or not), one or more rounded comers, and the like.
  • the elliptical blocks can have diameters (e.g., a diameter in a x-direction, a diameter in a y-direction), a parameter that describes the deviation from an ellipse or a rectangle (e.g., an elliptical exponent that is equal to 2 for an ellipse and increases the more the ellipsoidal cross section shape approaches a rectangle), and the like.
  • diameters e.g., a diameter in a x-direction, a diameter in a y-direction
  • a parameter that describes the deviation from an ellipse or a rectangle e.g., an elliptical exponent that is equal to 2 for an ellipse and increases the more the ellipsoidal cross section shape approaches a rectangle
  • the hypothetical layers and/or blocks of hypothetical layers can have various shapes of varying complexity.
  • the hypothetical layers and/or blocks of hypothetical layers can include combinations of shapes, such as a combination of rectangular shapes.
  • a library of hypothetical profiles and corresponding simulated diffraction signals are generated and stored for varying geometries and/or material as well as varying wavelength or angle of incidence based on the previously generated and stored blocks of hypothetical layers.
  • library 185 can be generated using a computer system 1600 .
  • the process of generating library 185 can involve performing a large number of complex calculations.
  • a set of hypothetical profiles is to be generated for a periodic grating that is to have a profile with a bottom CD of 200 nm.
  • a 10% process variation is expected for the bottom CD of the periodic grating, which means that the bottom CD is expected to vary between about 180 to about 220 nm.
  • the top CD is expected to vary between about 160 to about 180 nm.
  • the nominal thickness (i.e., the height) of the periodic grating is to be about 500 nm, and that a 10% process variation is expected, which means that the height can vary between about 450 to about 550 nm.
  • the desired resolution is 1 nm, which means that each parameter of the hypothetical profiles is varied by an increment of 1 nm.
  • the top CD of the hypothetical profiles is varied between 160 to 180 nm in steps of 1 nm.
  • the bottom CD of the hypothetical profiles is varied between 180 to 220 nm in steps of 1 nm.
  • the thickness/height of the hypothetical profiles is varied between 450 to 550 nm in steps of 1 nm.
  • each diffraction calculation uses 6 matrices with 9 orders and 8 bytes, which totals 17 kbytes. As such, in this example, to store the diffraction calculations for all of the 87,000 hypothetical profiles at 53 different wavelengths, a total of 78 Gigabytes is needed.
  • a greater number of simulated diffraction signals are stored in a library for structures having profiles that vary in more than one dimension than for structures having profiles that vary in just one dimension.
  • the computation time for a single solution for a structure having a profile that varies in one dimension scales as M 3 , where M is the number of retained harmonic orders in one direction.
  • the computation time for a single solution for a structure having a profile that varies on two dimensions scales as 8M 6 .
  • the difference in computation time for a single solution between the one-dimension structure and the two-dimension structure can increase by a factor of 10648.
  • computer system 1600 can include multiple processors 1602 configured to perform portions of the computations in parallel.
  • computer system 1600 can be configured with a single processor 1602 .
  • computer system 1600 can include a memory 1604 configured with a large amount of memory, such as 8 Gigabytes, 16 Gigabytes, 32Gigabytes, and the like, that can be accessed by the multiple processors 1602 . It should be recognized, however, computer system 1600 can be configured with any number and size of memories 1604 .
  • simulated diffraction signals for a set of hypothetical profiles to be stored in library 185 can be generated based on the diffraction calculations for blocks of hypothetical layers stored in cache 1006 . More particularly, for each hypothetical profile to be stored in library 185 , one or more blocks of hypothetical layers that characterize the hypothetical profile are selected from those stored in cache 1006 . Boundary conditions are then applied to generate the simulated diffraction signal for the hypothetical profile. The simulated diffraction signal and the hypothetical profile are then stored in library 185 .
  • the hypothetical profile can be stored in various formats, such as graphically, using the parameters that define the hypothetical profile, or both.
  • cache 1006 can reside in memory 1604 .
  • the blocks of hypothetical layers and hypothetical profiles stored in cache 1006 can be more quickly accessed by computer system 1600 , and more particularly processors 1602 , than if cache 1006 resided on a hard drive.
  • library 185 can reside on various computer-readable storage media.
  • library 185 can reside on a compact disk that is written to by computer system 1600 when library 185 is generated, and read by signal processing module 190 ( FIG. 1 ) when library 185 is used to determine the profile of periodic grating 145 ( FIG. 1 ) on wafer 140 ( FIG. 1 ).
  • one or more simulated diffraction signals are generated based on the previously generated and stored blocks of hypothetical layers.
  • the one or more simulated diffraction signals can be generated by one or more processors 1602 .
  • the blocks of hypothetical layers can be stored in cache 1006 .
  • the generated simulated diffraction signals can be stored in library 185 .

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US10/274,252 US7427521B2 (en) 2002-10-17 2002-10-17 Generating simulated diffraction signals for two-dimensional structures
JP2004545360A JP4805579B2 (ja) 2002-10-17 2003-10-14 二次元構造用のシミュレート回折信号の生成
KR1020057006261A KR101058476B1 (ko) 2002-10-17 2003-10-14 이차원 구조물에 대한 모의 회절 신호의 생성
PCT/US2003/032779 WO2004036142A2 (en) 2002-10-17 2003-10-14 Generating simulated diffraction signals for two-dimensional structures
AU2003279290A AU2003279290A1 (en) 2002-10-17 2003-10-14 Generating simulated diffraction signals for two-dimensional structures
DE10393515T DE10393515T5 (de) 2002-10-17 2003-10-14 Erzeugung von simulierten Beugungssignalen für zweidimensionale Strukturen
CNB200380101531XA CN100442067C (zh) 2002-10-17 2003-10-14 对二维结构生成模拟衍射信号
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